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MONTEF 



NPS-53Ac74121__. 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




A O(h^) Cubic Spline Collocation Method For 
Quasilinear Parabolic Equations 



D. A. Archer 



December 19 74 

Technical Report for Period 
September 1974-December 1974 



Approved for public release; distribution unlimited 



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D208.14/2:NPS-53Ac74121 



I for: 

stgraduate School, Monterey, California 93940 



NAVAL POSTGRADUATE SCHOOL 
Monterey, California 



Rear Admiral Isham Linder Jack R. Borsting 

Superintendent Provost 



The work reported herein was prepared for the Naval Postgraduate 
School, Monterey, California. 

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4 

A 0(h ) Cubic Spline Collocation Method for 
Quasilinear Parabolic Equations 



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D. A. Archer 



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Naval Postgraduate School 
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12/11/74 



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18. SUPPLEMENTARY NOTES 



19. KEY WORDS (Continue on reverse aide It neceaaary and Identify by block number) 

Collocation, Cubic Spline, Parabolic Equations 



20. ABSTRACT (Continue on reverse tide It neceaaary and Identify by block number) 

A modified version of the usual cubic spline collocation method is proposed 
and analyzed for quasilinear parabolic problems. Continuous time estimates of 
order O(h^) are obtained, via arguments based on certain discrete inner- 
products, for a uniform mesh and sufficiently smooth problems. .Two types of 
collocation at the boundary are studied and shown to yield 0(h) and 

7/2 
0(h ) rates of convergence. 



DD I JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE 



S/N 0102-014-6601 



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SECURITY CLASSIFICATION OF THIS PAGE (When Data Bnfrad) 



ACKNOWLEDGEMENTS 



I appreciate the advice and friendship of Prof. H. H. Rachf ord , Jr., 
who introduced me to collocation methods and directed the dissertation upon 
which this paper is based. I am also grateful to Prof. M . F. Wheeler, 
Prof. R. A. Tapia, and Dr. S. I. Chou for their interest and encouragement. 



1-1 



4 
A 0(h ) Cubic Spline Collocation Method for Quasilinear 

Parabolic Equations * 



1. Introduction . Consider the quasilinear parabolic equation 



(1.1a) c(x,t,u) u - u = f(x,t,u,u ) , 0<x<l, 0<t<T 

L XX X 

(1.1b) u(0,t) = b (t) , u(l,t) - b-CO , < t < T , 

■ o i 

(1.1c) u(x,0) = g(x) , < x < 1 . 



Let A = {0 = x < x < • • • < x = 1} be a partition of I = [0,1] , with 

I. = [x. .. x.l, h. =x. -x. . , and h = max {h.: 1 < i < N} . Then define 
l i-l, ill l-l l — — 



II, (J) = V: {V is a polynomial of degree < k on J} 



and 



IT, = {V: V e n, (I.) , 1 < i < n) . 
k,A k i — — 



For -1 <£< k-2 let 

s(A,k,£) = n k A n C l (I) 

be the space of piecewise polynomials of degree < k (order = k) on A with 

continuity I . Note that S(A,k,£) has dimension d[S(A,k,£)] = kN - (l+l) (N-l) 

In this paper we shall be primarily concerned with S, e S(A,4,2) , the usual 

h 

cubic spline space on A . 



This research was supported in part by the National Aeronautics and 
Space Administration (NASA). 



1-2 



Recently Douglas and Dupont [ 10, 11 ] have studied collocation procedures 
for (1.1) based on the spaces S(A,k,l) for k > 4 . Their main result is 
that collocation at the images of the k Gauss-Legendre points in each sub- 

interval I. yields uniform errors of order 0(h) and superconvergence 

2k- 4 
results at the knots {x. } of order 0(h ) if the solution of (1.1) 

k+2 
u £ H (I) . These estimates (but not the analysis) are essentially the 

same as those of deBoor and Swartz [ 7 ] for ordinary differential equations. 

The analysis of [11 ] is based on certain discrete innerproducts as is the 

analysis presented in this paper. 

Several authors [4 ,16 ,20 ] have studied collocation techniques for 

ordinary differential equations using smoother spaces S(A,k,£) with t > 2. 

k-2 
The general result obtained is that the convergence rate is 0(h ) , a 

suboptimal rate of convergence for such spaces. However, the procedures of 

Russell and Shampine [20 ] will provide 0(h) convergence (and superconvergence 

2k- 6 
at the knots of order 0(h )) for k > 6 if the collocation takes place at 

the images of the k-2 Lobatto points on each subinterval. Hence, it is ex- 
pected that these procedures can be extended to parabolic problems through 
careful mimicing of the arguments in [ 11]. These procedures will be studied 

in a later paper. 

2 
In [ 8 , 19] cubic spline methods with 0(h ) accuracy have been studied 

for linear versions of (1.1). Also, in [18 ] a cubic spline collocation pro- 
cedure for the heat equation has been proposed (but not analyzed) ; for a partic- 

4 2 
ular explicit time discretization 0(h + (At) ) convergence obtains. However, 

this procedure is essentially the standard explicit finite difference method 

for the heat equation and the high accuracy does not readily generalize to more 



1-3 



difficult problems or other time discretizations. 

In [ 2, 3] a variant of the usual cubic spline collocation method yielding 

4 
0(h ) convergence rates for nonlinear ordinary differential and quasilinear 

parabolic equations was studied. In this paper we describe the high-order pro- 
cedure and provide continuous time estimates for (1.1). Two types of boundary 

4 
collocation will be considered, yielding uniform estimates of 0(h ) and 

7/2 
0(h ) respectively. In a subsequent paper we shall investigate the effect 

of various boundary collocation techniques for high-order smooth spline approx- 
imations to (1.1). 

It should be noted that the particular approximation used here is essentially 
the same as that of Daniel and Swartz [ 9 ] for two point boundary value problems. 
The procedures were developed independently, the derivation of [2 ] preceeding 
that of [ 9 ]• The finite-difference method discussed by Hirsh [13 ] can 
be interpreted as a cubic spline method, and as such, it is quite similar to the 
present technique. 

This paper has four parts. In § 2 the basic notation of the paper is developed, 
Some discrete innerproducts for cubic splines are then defined and studied. The 
basic approximation technique used here is developed in § 3 from consideration 
of a simple two-point boundary value problem. In § 4 the main results are pre- 
sented (Theorem 4.1). 



2-1 



2. Notation . We shall use the standard notation [ 14] for L (I) spaces 

and Sobolev spaces H. (I). In particular, |v|j 9 = (v,v) with 

L Z (I) 

(f,g) = / f(x)g(x)dx . Also, H 1 (I) = {V e H I (I): v(0) = v(l) = 0} , 
J l ° 

with |v| , = | Dv I r, . Also, we use W (I) = {v: D v is abs. cont. 

it (i) i/d) 

o 

m oo 2 

< j < m-1 , D v e L (I)} with Ivl = Y Id v| 

J * % ' ''I'm i-t I I I I ao 

W (I) l<m L (I) 

The spaces L [0,T; X] are defined as usual for normed linear spaces X [ 14 ] . 



If v e L (I) is defined on A , then write 
Let | v| v = max v. . 

OO I I ^ ' ' „00 ' I I 

I 0<i<N 



v = v(x ) and v = (v ,v , "-.v ) 
l i ~ o 1 ' N y 



Define the difference operators Vv. = h. (v. - v. .) , Vv. = Vv.... , and 

r ill l-l i l+l ' 

2 — 
A v. = VVv. . In case v = v(x,y) , denote the differences with respect to a 

particular variable as 

(V v)( X;L ,y) = h. (v(x ,y) - v(x 1 _ 1> y)) ; etc. 



In the following A is uniform; i.e., A = {x. = ih : < i < N} . Then 

define the discrete innerproduct 

h N 
[v.w] = ^ I(Vl"i-l + V i W i ) 
i=l 

i N-l i 

with norm |v| = [v,v] . Also, let < v,w > = h I v.w. and |v| = < v,v > 

N 1=1 



Additionally, let < v,w ] = h £ v w. 

i=l X 



Recall the summation by parts formula: 



(2.1) < Vv,w > = - < v,Vw ] + v w - v w 



2-2 



The following results are easily established for cubic splines. Let 

S° = S u n H 1 (I) . 
h ho 



Lemma 2.1 If v,w e S, , then 
h 

2 4 2 

(2.2a) - < v",w > = (v\w') + lL(v",w")+ ^ (v' ! ' ,w" ' ) - j- B(v" ,w' ) , 

2 4 ' 2 

(2.2b) - < v" +^2 a2 v" ,w > = (v\w') + j^ (v"\w'") - ^2 B(v",w') , 

h 4 h 2 

(2.2c) [v\w«] = (v\w') - j2o (v" , ,w"') + ^2 {B(v'\w') + B(w'\v')} , 

.2 

(2. 2d) [v'\w»»] = (v",w") +j- (v"\w'") , 

where 

B(v,w) = v„w XT - v w 
N N 00 

Note that v,w e H (I) is not necessary for (2. 2d). 

o 

PROOF : We prove (2.2a-b); the remaining results are similar. Recall the 

corrected trapezoidal rule 



d 2 

C <Kx) dx = 5_£ [(j) ( c ) + ^( d) ]_ - 



12 



] + *J$~ * (4) CO , 5 e- (c,d) . 
c 



Applying this rule one interval at a time and summing yields 

2 
i 

12 



,4 2 

(v",w) ■ [v",w] ■■£- < Vv",Vw ] + ^ (v"\w'") - ^ B(v",w') 



for all v,w e S, . Summation by parts and v,w e H (I) imply that 

2 4 2 

(2.3) (v'\w) = [v",w] +~ < A 2 v",w > + ^ (v"',w'") - ^ B(v'\w') 



which is (2.2b). It is easy to show that 

(2.4) (v",w") = B(v",w') + < A 2 v",w > ; 
hence, (2.2a) follows from (2.3) and (2.4). 



2-3 



To apply these results, we need the inverse relations (not assumptions 
for splines). 

Lemma 2.2 [ 21 ] If v e II with h/ min h± < o , then 

l<i<N 

(2.5) |Dv| < Ch~ 1 ||v| , 

1 _ 1 

(2.6) | |v| | < Ch q p | |v| | , l<p<q<» . 

L q (I) L P (I) 

Let |v| = max |v(x)| . Then for v e S, , 
3 x-0,1 h 

(2.7) |b(v",v')| < 2 |v"| Jv'l < Ch" 2 Mvll 2 . 

8 3 H 1 (I) 

o 
Hence, by (2.2c) 

(2.8) |v| 2 + | v * | 2 < C| |v| | 2 , 

H 1 (I) 
o 

Since B(v ,, ,v') is not definite, the left sides of (2.2a) and (2.2b) 

(with v = w) are not norms equivalent to the H (I) norm on S, in general 

o n 

Of course, if v is periodic, B(v ,l ,v l ) = and the forms in question are 

actually equivalent to the H (I) norm. 

o 

It is not generally true that |v| and |v| ~ are equivalent; 

L Z (I) 

however, it is the case that 

(2.9) |v| < c| |v[ ' 
It is true that 



L 2 (I) 



(2.10) c, l|v!| 2 ? £ |v| 2 + h 5 |v"| 2 ,£C.J|v|| 2 2 

1 L Z (I) L Z (I) 



2-4 



For this note first of all that the exponent of h is correct by Lemma 2.2. 
Als °' if , ,2 .5, ..,2 



I 2 + h 5 |v"| 
then 



and 



v. =0 , < i < N 



V'' - , i = 0,N 



/ 1 ^ O \ 

It is then clear that v = . Hence, ||v| = liv| +h |v' ' | J 



is a norm 



on S. and (2.10) follows, 
h 



We shall also use the following notation 

< v,w > = < v + — A v , w > , 



for mesh functions v and w . 



3-1 

3. A Two-Point Boundary Value Problem . In this section we consider a cubic 
spline approximation to the two-point boundary value problem 
(3.1a) u" (x) = f(x) , x £ I 

(3.1b) u(0) = b Q , u(l) = b 1 . 

It is well-known 4 that collocation at the knots {x.}. n in 

l i=0 

S, ; i.e., finding U e S, such that 
h c h 

(3.2a) U" ( x .) = f(x.) , < i < N 

c 1 1 

(3.2b) U c (0) = b Q , U c (l) = b 1 , 

2 
has a convergence rate 0(h ) (and no better) in general. However, defining 

U e S u by 
n 

,2 

(3.3a) U"(x.) = f(x.) - -^r f"(x.) , 0<i<N 

l l 12 i 

(3.3b) U(0) = b Q , U(l) = b 1 

leads to the following results. 

Theorem 3.1 Suppose u e w (I) is the solution of (3.1) and U e S is 
defined by (3.3). Then the following estimates hold for e = u - U : 

(3.4a) llD^eM < Ch^MuM , < j < 3 . 

L (I) W°(I) 

The following super convergence results are also valid for 1 < i < N : 

(3.4b) |e.'| < Ch 4 ||u| | , 

1 W°(I) 

(3.4c) le! , I < Ch 4 I lul I , 

W 6 (I) 

00 

(3.4d) |e"(C.,)| < Ch 3 ||D 5 u|| ; j =1 , 2 , 

1J L (I.) 

(3.4e) |e'! , 1 I : Ch 2 I |d 5 u| | 

1 2 L (I.) 



3-2 



where 
(3.4f) x . = (x + x )/2 

1 — 5 1 — X 1 

and 
(3.4g) g = x . + (-1) J -^— , j - 1,2 . 

6+k 
Additionally, if u £ W (I) for < k < 2 , then 

(3.4h) U 1 ! + ^r (A 2 U"). = U'.' + 0[h 4+k ||D 6+ u|| ), fo 
1 12 X X \ L"tt)/ 



r 1 < i < N-l 
L~(I) ' 



Proof: Expand e' ' about x. , on I. to obtain for ^ T ^ h : 
l-l l 

(h 4 llf iv ll ) . 



+ h F 



Then it is straightforward that 



(3.6) | J e"(x)p 2 (x) dx| £ Ch 5 ||u| 



, , 1 < i < N , 

I. W (I.) 

1 1 



for any p e IT bounded independently of h . 



Let G (x;£) be the Green's function for v' ' = g on I subject to 
v(0) = v(l) = , and define G, (x ;£) = ( — G )(x;£) . Since 

l y 8x o/ 

G q (x; •) e W 1 (I) and G (x.; •) e 11. for < i < N , j = 0,1 , 



we have 



D j e(x.)| = |/g.(x. ; Oe M (0 d£ 
i '■£ J ± 

N /" 

i I L/Y x i s 

n=l I J 



C)e"(C) dC 
n 



(3 ' 7) 5 ? ,, ,, 

i ch D j; | mi 

n=l W (I ) 

n 

1 Ch |u| 

W 6 (I) 



10 



3-3 



The stability results of [ 22 ] and (3.7) imply 

(3.8) ||D J e|| < Ch 4_j | |u|| , < j : 3 . 

L (I) W (I) 



Integration of (3.5) from x. to x. j yields (3.4c). Estimates 

i — -L i — "2 

(3.4d-e) follow immediately from (3.5). 

Since for 1 < i < N-l and < k < 2 

2 

(3.9) (A 2 U"). = [A 2 (f -£- f ')] = f.' + o(h 2+k ||D 4+k f|| ) 
i 12 i x \ L - (I) / 

estimate (3.4h) is established, and the proof is complete. 



We now consider defining W e S, by (3.4h) neglecting the 

(4+k i i 6+k i i \ 
h D u ) terms. More precisely, define W e S. by 
L(I)' h 

h 2 
(3.10a) W"(x) = f(x) - — f"(x) , x = 0,1 

,2 

(3.10b) W|' + ^r (AW"). = f. , 1 < i < N-l 
l 12 7 i l 

(3.10c) W(0) = b , W(l) = b. . 



The following results then obtain . 

Corollary 3.2 Let u , U be as in Theorem 3.1. Define W e S, by 
(3.10). Let z = U-W and e=z+e=u-W. Then 

(3.11a) ||D J z|| Ch 4+k ||D 6+k u|| , < j < 2 . 

L (I) L (I) 

(3.11b) | Iz'" | | < Ch 3+k | |D 6+k u|| 

L (I) L (I) 

6+k 
if u e W (I) , < k < 2 . 

Furthermore, all the inequalities of (3.4) hold with e replacing e 



11 



3-4 



and the norm of u on the right side of each inequality being changed 



to 



w 6 (i) 



Proof : Equations (3.10a-b) yield an (N+l x N+l) linear system for W, 



(3.12) 
where 



A H" = f = f " |f (f ' ' '•••' ' f N , )T 



A = 



12 



12 
1 10 1 
1 10 1 











10 1 
12 



i i -1 1 i 3 
Since A is diagonally dominant with | |A | | _f_ — , W 1 ' (hence W) 

is uniquely defined by (3.11). By (3.4h) U' ' satisfies 



(3.14) 



A U" = f + 6 , 



where 



and 



i = (0,5 1 ,•••, $x-l' 0) 



. | ., 4+k | | 6+k 
6 . < Ch u 

l ' — ' ' 



, 1 < i < N-l 



L (I) 



Subtracting (3.12) from (3.14) yields 



(3.15) 



A z" = 6 



12 



3-5 



from which 

(3.16) |||z"||l - II*" | I < |||6|| < Ch 4+k ||D 6+k u|| 

I I loo l~ M r - 2 "~" r - ll L - (I) 



Since z is piecewise linear 

(3.17) ||«" | | < IN*"! I I ■= Ch 4+k ||D 6+k u|| 

L (I) ' L (I) 

Finally, the Green's function representation leads to for j = 0,1 

(3.18) |D J z(x)| = \f G (x ; Oz"(0 dc| £ ||G (x ; -)|| 

I J J L 



(I) L (I) 



which along with (3.17) establishes (3.11a). The remainder of the Corollary 
follows from homogeneity in h and the triangle inequality applied to 
e = z + e 



13 



4-1 



4_. Continuous Time Estimates . In this section we consider the continuous- 
in-time approximation to the solution of (1.1) by cubic spline methods. 
Define U : [0,T] -> S by 

2 
(4 ' la) ( U xx + 12 C(U)U xxt ) (x > t} = (\ (U)U t " V U)U xt " F h (U ' U x } ) (x ' t} ' X = °' 1 ' 

(4.1b) (c(U)U t - [U xx + \- 2 A 2 u xx ] )( x ., t) = (f(U,U x )) (x.,t) , l<i<H, 



(4.1c) U(0,t) = b (t), U(l,t) = b.(t) , 
o 1 



(4. Id) U(x,0) - g(x) - "small" , x e I . 

In (4.1a) we have used (supressing (x,t)) 



h 2 ? 
(4.2a) ^(40 - c(4>) - j^ D^ [c(<j>)] 



h 2 1 
(4.2b) B h (4>) = f- D£ [c(4»)J 



h 2 2 
(4.2c) F h (<M x ) - f(*,<j> x ) " ^2 D h t f ^'* x )J 



where 



(4. 2d) D^](x) = 



|(V x ^) 1 -|(V x ^) 2 , x=0 



! (V x *>N " 2 (V x *Vl ' X = X • 



14 



4-2 



and 



(4.2e) D h [<|,](x) = 



( 2( A 2 ^ - ( A 2 #) , x = 



2(A x^ N -i- (A x n^2- * = 1 



2+k 
Note that for ip e W (I) , < k < 2 

(4.3a) Rj 0|>) = D i|» - D* M = o(h k ||D k+ %|| ) 
n x h \ x L °°(I) / 

(4.3b) R^ ty) = D^ - D * [*] = O^llD^M M ) • 

In the following we shall assume that c and f are smooth functions of 
their arguments with c subject to the bounds 



(4.4) < m 5 c(x,t,<t>) <_ M < °° 

x e I , t e [0,T] and <J> e R = (-»,<») . 

The choice of this particular approximation is motivated by the results 

of §3, specifically Corollary 3.2. Later U(x,0) will be chosen to provide 

4 
the desired 0(h ) convergence rates. It is also possible to use a different 

collocation procedure at the boundary, namely 

(4.1a)' (c(U)U -U Wx,t) - f(x,t,U,U ) , x = 0,1 ; 

\ t XX / X 

7/2 
however, the analysis here will provide only 0(h ) rates of convergence. 

The analysis of (4.1) will proceed along the same lines as that in [ 11 ] 

and will employ the discrete inner products of §2 . Before beginning the 

error analysis, we establish the existence and uniqueness of the solution of 



15 



(4.3) 



(4.1). For this, we consider the equivalent matrix formulation based on the 
B-spline basis {V ,V ,...,V } on the knot set 



1 = t. (1< i < 4), t 4+ . - x. (1 £ i < N-l), x N+3+ . = 1 , (1 <_ i <_ 4) J; 



N+3 
see [5, 6 ]. Let U(x,t) = \ a.(t)V.(x) . Then (4.1) b 

j-1 J J 



ecomes 



(4.5a) £(oOa'(t) -^a ~J-(a) , t e (0,T] 

(4.5b) a x (t) = b Q (t) , a N+3 (t) = b^t) , t e (0,T] 



(4.5c) 



a(0) = given , 



where a(t) = (a 1 (t),a 2 (t) , . . . ,a (t)) , and for 1 < j < N+3 



C( I \ \ (x i ))V j (x i) 



1 < i < N-l 



(4.6a) [£,(a)] = 



t («: 



— c( £ a R V k (x i ))v!'(x i ) - A^C I a k V (x ± ))V (x £ ) 
k J k J 



+ B ( I a k V k (x i ))V'(x i ) , i = 0,N 
k J 



(4.6b) lj hi 



V!'( Xi ) +^ (A 2 V'.')(xi) , 1 < 1 < N-l 



-vj'( Xl ) 



, i = 0,N 



(4.6c) UCcOL 



it 



f( I \\( Xi ) , I a V»(x ± )) , 1 < i < N-l 



k k 



" F h ( ^ \ \ (x i } ' E \ K (x i- })ti = °' N 
k k 



16 



4-4 

The assumption that f is Lipschitz continuous with respect to its 
last two arguments implies that ^(ot) is likewise Lipschitz continuous. 
Thus, the local existence (in time) of the solution to (4.1) will be es- 
tablished in case 

(4.7) G?(a)£ = and Q ± = 6 N+3 = 

N+3 
implies that 3=0. Let cj) (x) = £ a, V (x) and suppose that 

k=l R R 

N+3 
iKx) = I B, V (x) with 8 satisfying (4.7). Then 
k=l k k 

(4.8a) ^( Xi ) =0 , < i < N , 

and 

(4.8b) {c(4>H + 2D^ [c(<J>)]ij> }(x.) =0 , i = 0,N . 

xx h x i 

By the standard cubic spline identities [15 ], (4.8) is equivalent to 



(4.9a) * xx (Vl } + ^xx (x i } + *xx (x i+l> = ° > 1 i ^ N-l , 

(4.9b) {c(<|>) - f- D* [c(<|>)]} (0)^(0) -^dJ [c(<|.)](0)i/; xx (x 1 ) =0 , 

(4.9c) {c(40 +Y»l [c(<j>)]}(D* xx (D +^D^ [c(«J.)](1)^ xx (x 1 ^ 1 ) = . 

Assuming that 

(4.10) \~- [c(x,t,<J>)]| L for x e I , t e (0, TJ, <j> £ R , 

dX 

we find from (4.3a) that {D. [c(<f>)]}(x) is bounded for x = 0,1 . Thus, 

h 

for sufficiently small h , (4.9) corresponds to a diagonally dominant, homo- 
geneous linear svstem for \b (x n - ) . Hence, \b '- and $ = . 

XX 1 XX ~ 



17 



4-5 



Lemma 4.1 If (4.10) holds then for h sufficiently small there exists a 
unique U e S, solving (4.1) for t e (0,T] . 



Lemma 4. 2 If 

(4.10a)' |-j~ [c(x,t,cj))]| £ L , x e I , t e (0,T] , <j> e R , 

and 

(4.10b)' |~ [f(x,t,*,V)]| £ L , x £ I , t £ (0,T], <j>,ijj e R, 

then for h sufficiently small, there exists a unique solution U, 6 S 

of (4.1) with (4.1a)' replacing (4.1a) for t £ (0,T] . 



Proof : Similar to the above. Just differentiate (4.1a)' with respect to t 

to obtain an analogue of (4.9) which is diagonally dominant for h small enough, 

We now turn to the convergence analysis of (4.1). Note that (4.1b) is 
equivalent to the discrete Galerkin formulation 

(4.11) < c(U)U ,V > - < U ,V > = < f(U,U ),V > , V € S, . 
t xx A x h 

For the analysis define the comparison function W . [0, T] ■> S, by 

h 2 

(4.12a) W (x,t) =u (x,t) - — -u (x,t) , x = 0,l , 
xx xx 12 xxxx 

h 2 2 
(4.12b) CW^ + yj ^WJdi.t) -u^Cxi.t) , 1 < i < N-l , 

(4.12c) W(0,t) = b Q (t) , W(l,t) = b 1 (t) 

4 
Note that Corollary 3.2 implies that W (W ) is a 0(h ) approximation to 

u (u t ) . 



18 



4-6 



Let z = W - U £ S , e=u-W, and e=z+e=u-U . Our 



h 
plan is to estimate z in terms of e and then to bound e using the bounds 

on z and the triangle inequality. In the following analysis, we shall often 

2 2 

require the inequality ab < ea + (l/4e)b for a,b :1 , any e > . 

From (4.12b) and (1.1) we find 

< c(W)W ,V >-< W ,V > = - < c(W) e ,V > 
t xx A t 



(4.13) 



+ < [c(W) - c(u)]u .V > + < f(u,u ),V > , V e S, 

t x h 



Subtract (4.11) from (4.13) and apply the assumed smoothness of c and 

f to obtain 

< c(U)z ,V > - < z ,V > = < [c(W) - c(U)]W ,V > - < c(W)e ,V > 

+ < [c(W) - c(u)]u t ,V > 

+ < f(u,u ) - f(U,U ), V > 

X X 

(4.14) = < c zW ,V > - < c(W)e ,V > 

u t t 

* ~ 
- < c e u ,V > . 
u t 

a ~ a ~> 

+ <fe+f e,V> 

u U X 

X 

+ <fz + f z,V> , 

U U X 

X 



where the partial derivatives c ,f ,f are evaluated as required by the mean 

u u u 
x 

value theorem. Now use Cauchy-Schwarz, the boundedness of the derivatives of 
c and f , and the trivial inequality mentioned above with V = z : 

2 

(4.15) < c(U)z .z > - < z ,z > < C(|z|' : + |z l + I Te | ) + 6 I z^ ' , 5 > 
t t xx t A x 1 t 1 



19 



4-7 



Here 


(A. 


16) 


If 




(4. 


17) 



~,2 ,~|2 ,~ ,2 ,~ ,2 

Te = e + e + e I 
i ii i x i i 1 1 



u,u t e L 2 [0,T;W 6 (I)] , 



Corollary 3.2 Implies that 

(4.17b) / |Te| 2 dx < Ch 8 (| |u| | 2 fi + ||u|| 2 fi ) . 
V l/[0,T;W°(I)] t l/[0,T;W°(I)] 

Choose 6 so that n = m-6 > and use (2.8) to obtain 

(4.18) n|z.| 2 - < z ,z > A lC(||z|| 2 +|Te| 2 ) . 
t xx t A H l 

o 
To complete the estimate, it is necessary to consider the boundary terms 

(4.1a). A straight-forward computation using (4.12a) and (4.3) yields 



h 2 



{W + — - c(W)W )(x,t) = {A. (u)u - B, (u)u - F, (u,u )}(x,t) 
xx 12 xxt h h -x-t- h x 

h : 



xt 
{c(u)e xxt }(x,t) 



(4.19) 

+ R^x.t) , x = 0,1 , 

where 

2 

(4.20a) R^x.t) = - j2 {R ^ [ c ( u ^ ~ 2R J t°( u )J " ^ [f (u,u x ) ] } (x, t) 



No 



te that if I h = [0,x 3 ] u [^ ,1] 



(4.20b) c(x,t) = c(x,t,u) € W 4 ^) 

and 

(4.20c) I(x,t) = f(x,t,u,u ) e W 4 (I ) , t e (0,TJ 

x n 

then by (4.3) 

(4.20d) l\l 3 Wl Ch 4 (||D*7 ll L . (Ih) + ll»?ll L . ) Ct) 

20 



4-8 



In the sequel, we assume that 

(4.20e) K. = | |d 4 7| I + I Id 4 71 1 < oo . 

L [0,T;L (I h )] x L [0,T;L (1^) ] 

Subtract (4.1a) from (4.19) to obtain 

h 2 
Z xx + l2 C(u)z xxt = K (u) " \ (U) K 

" ( B h (u) " B h (U) Kt 

" (V U) (Z xt + •«?) 

(4.21) - (F, (u,u ) - F, (u,U )) 

h x h x 

" <V U 'V " F h (U ' U x» 

h 2 

- 12 C(U) e xxt + \ 

We now estimate the terms on the right side of (4.21). The treatment of all 
but one of the terms is somewhat rough. 

- I^(l D h [c(u) - C( W]I 3 + l D h [c(w) " c(u) l 3 ) 

* Cdllelll + IIUIIL) 



B, (u) - B, (U) L < Ch(| le | + z ) 

h h ' 3 — M ' x M 'o° i i i x i i i w / 

F, (u,u ) - F,(u,U )L ■: C(|||e ||| + ||z ||| ) 
h x hx'3— '" x 1 "" i i i x i i I* 



F h (u,U x ) - P h (U,U x )| < C(|||e||| w + IIWHJ • 



21 



Similarly, 


(4. 


22b) 


(4. 


22c) 


anc 


I 


(4. 


22d) 



4-9 



Recalling (4.10) and (4.3a) with k = , 

l B h (U) I a < Ch 2 • 
Thus, 



(4.22.) IV^xt + ^U^Kt^ l«xtU ) 



Now multiply (4.21) by z , integrate in t , and apply (4.22) 



to obtain 



-t 

4 |« | 2 (t)+yh 2 f|z J 2 dx < Ch" 2 J ( |||z| || 2 + |||z Ml 2 + h 4 |z 
2 ' xx '3 J ' xxt ' 9 — o ^ IMII| oo i i i x i i i M i 



2 
xt'9 




(4.23) ^ + |T^| 2 + l^l^dt 

+ \ KA^ > 

where y > 

(4.24a) |Te| 2 = | | |e| | | 2 + | | |e | | | 2 + h 4 (|e I 2 + |e J 2 

I 13 I I I I I loo I 1 I x l I loo \' xt'3 ' xxt '5 

(4.2440 f |Te| 2 dx ■: Ch 8 / ( | |u| | 2 , + | | u I | 2 , \ dx , if (4.17a) holds, 

3 > W b (I) fc W b (I)/ 

It is clear from (4.21) and the bounds (4.22) that 

2 

l z L TT- l z L + C(| I |'z| I I + I I |z I I I +h 2 |z L 

1 xx 1 3— 12 ' xxt '8 \ MI ' M °° 1 1 1 x i 1 loo ' xt'3 



(4.25) 



l\la) 



+ l Te l 3 + KU) » t e (o,t] . 



Use of (4.23) and (4.25) permits the completion of the estimate (4.18). 

From (2.2b) , 

1 _d_ / I I M 2 _h_ ,, I ,2 

' Z xx' Z t A " 2 dt \ ' |Z| ! H^(I) 180 ' ^xxx 1 ^2 

(4.26) h 2 _, . 

"12 B(z xx' Z xt ) * 



22 



4-10 



Observing (via Lemma 2.2) that for any e > 



fJ|B( Zxx , Zxt )li.eh 3 | Zxt |2 + ch | Zxx ,2 



(4.27) 



" £ Nz t M 2 2 + Ch|z \ 2 , 
t L Z (I) xx 9 



and adding (4.27) to both sides of (4.18) yields 

4 



t i 2 4i(iNi; V) ^n^ii; 2(l) ) 



l\\ II 2 
(INI ! 

\ H i 



+ h|z | 2 + | Te| 2 
, . xx a 

o 



+ -ll- t l| 2 2 

L (I) 

I I 2 
Integrate (4.28) with respect to t , apply (4.25) to bound the h|z | 

XX o 

term, and apply Lemma 2.2 to produce 

2 



n/ |i J 2 dx +|||z|| 2 (t) 1 C S {\\z\\\ + 
fc 2 1^(1) ° l H 



Te 



(A. 29) 



+ h(|Te| 2 + |Hj*>}dT 



5 t 2 

+ Kh / I z L dx 
' xxt ' 3 



+ (e + K h 2 ) f\ \z\ | 2 dx 

V /0 fc L Z (I) 



+ c | |z|| 2 (0) . 

H (I) 
o 



23 



4-11 



5 f t i 1 2 

(n + K)h I |z J di to (4.29) using the estimate 



Now add 

xxt ' 3 



of (4.23) . Thus, proceeding as in (4.29) 



j; 



2 . . !,,_,, 2 -"",11* + ItS 



n/ MI..IM 2 dT + I||z|| 2 (t) <cj\, 

o c i r(i) t) I h x (d 

o o 



+ h(|le|^ + 1^1*) j 



dT 



(4-30) , „*,2. /^ | | I 1 2 

+ (e + K*h ) I z dx 

^0 c l 2 (I) 



+ C llzH 2 , (0) 

H X (I) 
o 



24 



4-12 



By (2.10) and Gronwall's inequality with e and h sufficiently small, 
we obtain our basic estimate 



H z tH 2 2 2 + I!»M 2 - 1 

l/[0,T;l/(I)] L [0,T;H^(I)] 

<*■»> £ c/jlTel^htlTel^lRj^ 

+ C||z|| 2 (0) . 

IT (I) 

o 

If (1.1) is smooth enough that (4.17a) and (4.20e) hold, then the use of 
(4.17b), (4.20d), and (4.24b) in (4.31) implies the estimate 

MM 2 II I I 2 

l/[0,T;l/(I)] L°°[0,T;h\i)] 



^0 ' u t' 



2 



(4 - 32) " \ t L 2 [0>T;W 6 (I) j l 2 [0,T;W 6 (I)] 

+ hK?) + cllzH 2 . (0) . 

^> V(I) 

o 

There are a variety of interpolation schemes which produce U(x,0) such 

that |z| -, (0) = 0(h ) . See [15] for several possible choices. Here we 

o (I) 
choose the interpolation studied in [9]; namely, define U(x,0) by 



(4.33a) U(x.,0) = g(x.) , < i < N 

(4.33b) U (x,0) = g"(x) - ^rg (iv) (x) , x-0,1. 

xx 12 



It is then easy to see (compare with Corollary 3.2) that if g e W (I) 

(4.34) ||z || (0) £Ch 4 ||g|| 

XX L (I) W (I) 



25 



4-13 



Our final estimate is then 



(4.35) ||i || 2 2 + | |z|| M x £Ch 4 (||,u || 2 6 

C l/[0,T;l/(I)] L [O t T;H:(D] V l/[0,T;W°(I) ] 



+ I sl I f, + \ ) » 
w 6 (i) N 

2 
after using the fact that for <J> e L [0,T;X] , <J>(0) e X , 

IMI 2 2 < c IU t ll 2 2 + 1 1*1 lx (0) • 

l/[0,T;X] IT[0,T;X] 

00 6 

The further assumption that u e L [0,T;W (I) J , and Corollary 3.2 imply that 
(4.36) I lei I Ch 4 ||u|| , 

x ■ ' ' I ' ' 00 oo — II I' 00 h 

L [0,T;L (I)] L [0,T;W (I)] 

Using (4.35), (4.36), the triangle inequality and the embedding of H (I) in 

o 

L°°(I) ; i.e., | <J) | .S C | | <J> | i » <J> e H (I) , we obtain the uniform 

L°°(I) ' H X (I) ° 

o 
estimate 



e | | 1 Ch 4 (, , 

L ro [0,T;L°°(I)] L°°[0,T;W b (I)] 



llu 

l iu,i;l, u;j 

(4.37) 



■\)' 



+ lu.l 2 , + 

l/[0,T;W D (I)] 



Theorem 4.1 Suppose that c, c , f, f , f are uniformly bounded mde- 

rr u u u 

x 

pendently of their arguments and that (4.10) and (4.20b,c) hold. Then for 
h sufficiently small there exists unique U solving (4.1) and (4.33). If 
u , the solution of (1.1), satisfies 

(4.38) ue L a5 [0,T;W 6 (I)] , ty L 2 [0,T;W 6 (I) ] , 
then 

(4.39) | |u-U| | 1 Ch 4 ( | |u| | 6 

L"[Q,'T;i."(I)] ' L°[0,T;W (I)] 



+ M u t ll 2 6 +K h) * 



L [0,T;W"(I)] 



9A 



4-14 



We now show that the use of the simpler boundary collocation (4.1a)' 
yields 0(h ) estimates; suboptimal in L (I) or L (I) norms, but 
optimal in H (I) . This loss of accuracy is not believed to be actual; 
rather, it is just a function of the particular analysis employed. We 
proceed as before with U defined by (4.1), (4.1a) 1 and W by (4.12). 
Then the error analysis is unchanged through (4.18). Note that for x = 0,1 



(4.41) c(W)W^ - W = f(u,u ) + e 

t XX X XX 



where 

h 2 
(4.42) e (x,t) = — u (x,t) , x = 0,1 . 
xx 12 xxxx 

Subtract (4.1a)' from (4.41) and use the boundary values to find 



(4.43) 



-z = f(u,u ) - f(U,U ) + e 

XX X X XX 



* ~ ~ 

= f (z + e ) + e 

u x x xx 

X 



Thus, 

- B(z ,z J = B(f z ,z J + B(f e + e ,z ) 

XX Xt u x xt U X XX Xt 
X X 

(4.44) 

1 * d 2 * ~ ~ s 
= ^ B(f z ,— z ) + B(f e + e ,z ) 

2 u x dt x u x xx xt 

x x 

Integrate by parts 

-f'BU ,z ) dT .-A(*,(L.( £ * ).z 2 )dT + -iB(f\*Mr 

J Q xx' xt y 2 J \ 3t \ u x / x / 2 \ u x x / | o 

(4.45) -f b(|- (f* e +e )z )di 

' J n V 3t \ U X XX n X / 

u X 

I t 

+B(t e+e,z) 

\ u X XX x / I o 



27 



4-15 



Now assume that 



,3 2 f 
(4,A6) Iftlu" CXft,*,*>| £L<» , x £ I , t e [0,T],<M e R. 

x 

Then 

Jr fc i -1 f t i .2 f fc i ~i2 

! B(z ,z Jdi (C + eh ) L z L di + Ch J Te ' di 

xx xt ' — '•'O ' x'9 *0 ^ 

(4.47) + (C + eV 1 ) |z |*(t) + Ch|Re|^ (t) 

X d a 

+ C(h _1 |z | 2 + h|Re|^ )(0) , 

X o d 

where, 



i~i2 ,~i2 ,~i2 i~i2 i~ i 2 

(4.48a) Te r = e r + U U + e \Z + e J* 

1 '5 'x'9 ' xx '9 I xt'9 ' xxt'9 

and 

(4.48b) \Re\l = \e \ 2 + |e | * 
1 '9 'x'9 ' xx ' 9 

Integrating (4.18) with respect to t and applying (4.26) and (4.46) yields 

/t 2 1 i i i i2 1**711112 

z I dx + - z (t) < C j .(. z - 

E 1 (I) V h 1 ! 

) 



+ ItSI 2 



(I) H (I) 

o o 



3. ~i 2 
h iTelg 1 

r 

+ (Ch + eh) Jjzjg di 

* 2 * 1 2 

+ (C h + e h)|z | (t) 

X o 

+ Ch 3 |Re|^ (t) 

+ c/h|z + h 3 |Ref' 



+ z 



x'9 ' ' 9 

2 



i, >> 



H <I) 



28 



4-16 



Apply Lemma 2.2 several times and take h so small that ( ^- C h - e ) >0 
Then 

/' l z t | 2 dx + ||z|| 2 (t) < c/'dlzH 2 . + IT^I 2 + h 3 |Te| 2 )dT 



H (I) H'd) 



(4.50) 



+ c(h 3 ( max |Re| 2 )+ ||z|| 2 (0) ) 



0<x<t H (I) 

o 



Gronwall's inequality implies that 



J |z I dT + |Z, 

L [0,T;h'(I)] 



(4.51) icf ( |Te| 2 + h 3 |Te| 2 )di 

d 

:(h 3 ( max |Re| 2 )+ ||z|| 2 (0) ) 
V n<t-<T d p-v-n ' 



+ C 



0<t<T H (I) 

o 



Note that 



// con u 3 l I 2 h i I 2 , 3i~ i2 h | i2 
(4.52) he I —7-7 iu ; h e L = -- ,, u 

1 xx '3 144 ' xxxx'3 ' xxt ' 3 144 ' xxxxt ' 3 



Hence, under assumption (4.17a) and the choice of U(x,0) given in (4.33), we 

obtain 

T 

2 , 1,1,2 



f 



L°°[0,T;H^(I)] 



(4.53) ± Ch 8 (||u t || 2 2 6 + ||g|| 2 6 ) 

K } t L Z [0,T;W b (I)] W°(I)> 



+ Ch (J, Iu L d: + max Iu ) 
y ' xxxxt '3 n „ ' xxxx ' 3 / 



29 



4-17 



3 
This estimate leads easily to the optimal 0(h ) estimate in the 

1 7/2 2 

H (I) norm; however, it implies only 0(h ) estimate in the L (I) 

oo 

or L (I) norm. Of course if 



(4.54) u (x,t) = , x = 0,1 , t e [0,T] , 

4 
the estimate becomes 0(h ) . Similarly, if u is periodic, i.e., 

(4.55) D^ u(0,t) = D^ u(l,t) , £ j £ 2 , t e [0,T] , 

4 
then making U similarly periodic results in 0(h ) estimates since 

B(z , z ) = 0. Note that in this case (4.1a) is replaced by (4.55) as 
xx xt 

applied to U . 



Theorem 4.2 Under the assumptions of Theorem 4.1 , with (4.10)' and 
(4.46) replacing (4.10) and (4.20), let U be defined by (4.1) with (4.1a)' 
replacing (4.1a). Then, 



(4.56a) ||u-u|| , lCh 3 (||u|| , + | |u | | 2 , 

L°°[0,T;H o (I)] V L°°[0,T;W (I)] L [0,T;W°(I)] 

(4.56b) II U -U|I LOO[0>T;L - (I)1 iCh 4 (||u|| m 6 + Mu t ll 2 6 

L IU,i,l u;j \ L [o,T;W°(D] t L [0,T;W°(I)] 



) 



7/2/ , i /f T , i 2 \ l/2 \ 

+ Ch ' ( max u L+jl u „L 

\0<t<T XXXX 3 Vo xxxxt ' 3 / / 



If, in addition, (4.54) holds 

(4.56c) ||u-u|| <_ Ch ( | ] u | | , + ||u || _ , 

L"[0,T';L"(I)i " V L°°[0,T;W b (I)] t L Z [0,T;W b (I) ]. 

Furthermore, if (4.55) holds, and U is also required to be periodic, then 

(4.56c) obtains. 

30 



K-l 



.REFERENCES 

1. E.L.Albasiny and W.D.Hoskins, Increased accuracy cubic spline 
solutions to two-point boundary value problems, J . Inst .Maths .Applies . , 
9(1972), 47-55. 

2. D.A.Archer, Cubic spline collocation methods for nonlinear parabolic 
problems, contributed:Fall S I AM-SIGNUM meeting, Austin, Tx. , Oct. 1972. 

3. , Some collocation methods for differential equations, Ph.D. 

Thesis, Rice Univ., Houston, Tx. , 1973. 

4. C.R.deBoor, The method of projections as applied to the numerical 
solution of two point boundary value problems using cubic splines, 
Ph.D. Thesis, Univ. of Michigan, Ann Arbor, 1966. 

5. , On calculating with B-splines, J.Approx. Thy • ,6(1972) , 

50-62. 

6. , Package for calculating with B-splines, to appear SIAM J. 

Num. Anal. 

7. , B.K.Swartz, Collocation at Gaussian points, SIAM J. Num. Anal., 

10(1972), 582-606. 

8. J. C. Cavendish, Collocation methods for elliptic and parabolic boundarv 
value problems, Ph.D. Thesis, Univ. of Pittsburg, 1972. 

9. J.W.Dainel and B.K.Swartz, Extrapolated collocation for two-point 
boundary value probelms using cubic splines, Los Alamos Scientific 
Laboratory Technical Report LA-DC-72-1520 , Dec. 1972. 

31 



R-2 



10. J.Douglas , Jr. , and T.Dupont, A finite element collocation method for 
quasilinear parabolic equations, Math. Comp., 27(1973), 17-28. 

11. , , Collocation methods for parabolic equations in a 

single space variable, Springer-Verlag , Berlin, 1974. 

12. D.J.Fyfe, The use of cubic splines in the solution of two-point 
boundary value problems, Computer Journal, 12(1969), 188-192. 

13. R.S.Hirsh, Application of a fourth order differencing technique to 
fluid mechanics problems, contributed: Fall SIAM meeting, Alexandria, 
Va. , Oct. 1974. 

14.' J. L. Lions and E.Magenes, Non-homogeneous boundary value probelms 
and applications, Springer-Verlag, New York, 1972. 

15. T.R.Lucas, Error bounds for interpolating cubic splines under various 
end conditions, SIAM J. Num. Anal, , 11(1974) ,569-584. 

16. , G.W.Reddien, Some collocation methods for nonlinear 

boundary value problems, SIAM J. Num. Anal., 9 (1972) ,341-356 

17. , , A high order projection method for nonlinear 

two point boundary value problems, Numer.Math. , 20(1973) ,257-270. 

18. N.Papamichael and J.R.Whiteman, A cubic spline technique for the one 
dimensional heat conduction problem, J . Inst .Maths. Applies . , 11(1973), 
111-113. 



32 



R-3 



19. S.G.Rubin and R.A.Graves, A cubic spline approximation for problems 

in fluid mechanics, School of Engineering Old Dominion Univ. Technical 
Report 74-T1, Norfolk ,Va. ,June 1974. 

20. R.D.Russel and L.F. Shampine , A collocation method for boundary value 
problems, Numer.Math., 19(1972) , 1-28. 



k-i k 
21. B.K.Swartz, 0(h oj(D f»h)) bounds on some spline interpolation 

errors, Los Alamos Scientific Laboratory Technical Report, LA-4477,1970 



22. , R.S.Varga, Error bounds for spline and L-spline interpolation, 

J.Approx.Thy. , 6 (1972) ,6-49 . 



33 



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Houstan, Texas 77001 

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Department of Mathematics 
University of Chicago 
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Madison, Wisconsin 53706 

Professor G. Fairweather 
Department of Mathematics 
University of Kentucky 
Lexington, Kentucky 40500 

Professor R. D. Russell 
Department of Mathematics 
Simar Frazer University 
Bunaby 2, B. C 
Canada 



34 



Professor J. Daniel 1 

Department of Mathematics 
University of Texas - Austin 
Austin, Texas 78700 

Dr. L. F. Shampine 1 

Sandia Laboratories 
Albuquerque, New Mexico 87100 

Professor G. W. Reddien 1 

Department of Mathematics 
Vanderbilt University 
Nashville, Tennessee 37200 

Professor L. Wahlbin 1 

Department of Mathematics 
Cornell University 
Ithaca, New York 14850 

Dr. M. Ciment 1 

Naval Surface Weapons Center 

White Oak Laboratory 

Silver Spring, Maryland 20910 

Dr. J. Enig 1 

Naval Surface Weapons Center 

White Oak Laboratory 

Silver Spring, Maryland 20910 

Professor C. Comstock 1 

Department of Mathematics 
Naval Postgraduate School 
Monterey, California 93940 

Professor R. A. Franke 1 

Department of Mathematics 
Naval Postgraduate School 
Monterey, California 93940 

Professor D. A. Archer 15 

Department of Mathematics 
Naval Postgraduate School 
Monterey, California 93940 



35 



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DUDLEY KNOX LIBRARY - RESEARCH REPORTS 



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